Respuesta :
Answer:
3) cos 75 = (√6 - √4)/4
4) sin 105 = (√6 + √4)/4
5) below
6) θ = 39.2° or 140.7°
7) θ = 120° or 240°
8) θ = 0.267π rad or π rad
Step-by-step explanation:
3) cos 75 = cos (30+45) = cos30.cos45 - sin30sin45=
√3/2 . √2/2 - 1/2 . √2/2 = (√6 - √4)/4
4)sin 105 = sin (60+45) = sin60.cos45 + sin45.cos60 =
√3/2 . √2/2 + 1/2 . √2/2 = (√6 + √4)/4
5)sin4θ = 4sinθcosθcos2θ
sin4θ = sin(2θ+2θ) = sin2θcos2θ + sin2θcos2θ = 2sin2θcos2θ
As sin2θ = sinθcosθ + sinθcosθ = 2sinθcosθ
substituting:
2.2sinθcosθ.cos2θ = 4sinθcosθcos2θ
6) 5cos2θ = 1 0°<θ<360°
cos2θ = 1/5
As cos2θ = cosθcosθ - sinθsinθ = cos²θ - sin²θ
and sin²θ + cos²θ = 1 → sin²θ = 1- cos²θ
we can say that cos2θ = cos²θ - (1- cos²θ) = cos²θ - 1 + cos²θ = 2cos²θ -1
So, 2cos²θ - 1 = 1/5
2cos²θ = 1/5 + 1
2cos²θ = 6/5
cos²θ = 6/10
cos²θ = 3/5
cosθ = ± √(3/5)
θ = +cos⁻¹√(3/5) → θ = 39.2°
θ = -cos⁻¹√(3/5) → θ = 140.7°
7) cos2θ = cosθ 0°<θ<360°
From the item above, we know that cos2θ = 2cos²θ - 1
2cos²θ - 1 = cosθ
2cos²θ - cosθ - 1 = 0
Making cosθ = y to facilitate:
2y² - y - 1 = 0
Δ = (-1)² - 4.2.(-1) = 9
√Δ = 3
y = (1±3)/4
y₁ = 4/4 = 1
y₂ = -2/4 = -1/2
cosθ = y
cosθ = 1 → θ = 0°
cosθ = -1/2 → θ = 120° or 240°
As 0°<θ<360° (no equal sign) → θ = 120° or 240°
8) cos2θ = cosθ + 2 for 0 < θ < 2p
From the item above, we know that cos2θ = 2cos²θ - 1
2cos²θ - 1 = cosθ + 2
2cos²θ - cosθ - 3 = 0
Making cosθ = y to facilitate:
2y² - y - 3 = 0
Δ = (-1)² - 4.2.(-3) = 25
√Δ = 5
y = (1±5)/4
y₁ = 6/4 = 2/3
y₂ = -4/4 = -1
cosθ = y
cosθ = 2/3 → θ = 48.2° = 0.267π rad
cosθ = -1 → θ = 180° = π rad
